Eigen-Degeneracy as a Geometric Proxy for Malignant Transformation and Therapy Resistance: A Mathematical Oncology Framework

We propose that malignant transformation, therapy resistance, and relapse can be characterised geometrically as eigen-degeneracy: a collapse of dynamical separation and local stability margins in operators governing cell state dynamics. Operationally, degeneracy appears as crowding of slow modes (loss of timescale separation) and, in non-normal dynamics, as a shrinking pseudospectral stability radius that permits large transient amplifications even when eigenvalues indicate asymptotic stability.

This framework unifies the cancer attractor hypothesis with critical transition theory, and provides three computable metrics from state-resolved single-cell assays and spatial transcriptomics: Jacobian-based degeneracy DJ (slow-mode multiplicity and margin, inferred from RNA velocity and vector-field methods), pseudospectral degeneracy Dε (robust distance-to-instability under perturbations, capturing non-normal transient growth), and covariance-based degeneracy DC (effective dimension of local fluctuations on a latent state manifold, via participation ratio).

However, covariance-based estimates from snapshot data can be sensitive to compositional confounding: if experimental conditions induce large shifts in the proportions of major dynamical programmes (notably cell-cycle arrest), global comparisons of DC may reflect mixture changes rather than within-state geometry. Practical implementations should therefore quantify composition, report phase- or programme-matched comparisons where needed, and treat DC as a screening proxy rather than a universal classifier.

The central hypothesis is that degeneracy is a computable proxy for plasticity, understood as increased susceptibility to noise-driven switching among phenotypic states that are already available to the system. We derive falsifiable predictions: (1) baseline degeneracy predicts emergence of drug-tolerant persisters; (2) degeneracy peaks at microenvironmental stress niches; (3) longitudinal increases in degeneracy precede conventional tumour-burden signals in cohorts where repeated state-resolved sampling is feasible.

The framework suggests therapy design principles: effective strategies must not only kill malignant cells but also suppress phenotype switching by increasing stability margins and restoring timescale separation in healthy basins.

Status: Preprint. Preliminary empirical validation (8 datasets, ~100,000 cells) included; independent replication invited.